The reader is referred to [1]} for details, which we will not need in the following. To finish this subsection, we quote the Hamiltonian that is obtained in the above process in the case of four scalar fields.
The Hamiltonian \(H\)  takes the form
\(H_{\rho _l,\rho _r} = - \frac{\partial ^2}{\partial u^2_1} - \frac{\partial ^2}{\partial u^2_2} + V^{\rm CS}_{\rho _l,\rho _r}(u_1,u_2),\) 
